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Holt Algebra Variation Functions 8-1 Variation Functions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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Holt Algebra Variation Functions Warm Up Solve each equation x = 1.8(24.8) = x9 2 Determine whether each data set could represent a linear function. x2468 y12643 x–2–101 y–6– no yes

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Holt Algebra Variation Functions Solve problems involving direct, inverse, joint, and combined variation. Objective

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Holt Algebra Variation Functions direct variation constant of variation joint variation inverse variation combined variation Vocabulary

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Holt Algebra Variation Functions In Chapter 2, you studied many types of linear functions. One special type of linear function is called direct variation. A direct variation is a relationship between two variables x and y that can be written in the form y = kx, where k 0. In this relationship, k is the constant of variation. For the equation y = kx, y varies directly as x.

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Holt Algebra Variation Functions A direct variation equation is a linear equation in the form y = mx + b, where b = 0 and the constant of variation k is the slope. Because b = 0, the graph of a direct variation always passes through the origin.

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Holt Algebra Variation Functions Given: y varies directly as x, and y = 27 when x = 6. Write and graph the direct variation function. Example 1: Writing and Graphing Direct Variation y varies directly as x. Substitute 27 for y and 6 for x. Solve for the constant of variation k. Write the variation function by using the value of k. y = kx 27 = k(6) k = 4.5 y = 4.5x

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Holt Algebra Variation Functions Check Substitute the original values of x and y into the equation. Example 1 Continued y = 4.5x (6) 27 Graph the direct variation function. The y-intercept is 0, and the slope is 4.5.

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Holt Algebra Variation Functions If k is positive in a direct variation, the value of y increases as the value of x increases. Helpful Hint

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Holt Algebra Variation Functions Check It Out! Example 1 Given: y varies directly as x, and y = 6.5 when x = 13. Write and graph the direct variation function. y varies directly as x. Substitute 6.5 for y and 13 for x. Solve for the constant of variation k. Write the variation function by using the value of k. y = kx 6.5 = k(13) k = 0.5 y = 0.5x

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Holt Algebra Variation Functions Check Substitute the original values of x and y into the equation. y = 0.5x (13) 6.5 Check It Out! Example 1 Continued Graph the direct variation function. The y-intercept is 0, and the slope is 0.5.

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Holt Algebra Variation Functions When you want to find specific values in a direct variation problem, you can solve for k and then use substitution or you can use the proportion derived below.

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Holt Algebra Variation Functions The cost of an item in euros e varies directly as the cost of the item in dollars d, and e = 3.85 euros when d = $5.00. Find d when e = euros. Example 2: Solving Direct Variation Problems Substitute. Solve for k. Use 0.77 for k. Substitute for e. Solve for d. e = kd Method 1 Find k = k(5.00) 0.77 = k Write the variation function = 0.77d d e = 0.77d

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Holt Algebra Variation Functions Substitute. Find the cross products. Method 2 Use a proportion. e1e1 = d1d1 d2d2 e2e = 5.00d d = Solve for d d Example 2 Continued

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Holt Algebra Variation Functions The perimeter P of a regular dodecagon varies directly as the side length s, and P = 18 in. when s = 1.5 in. Find s when P = 75 in. Check It Out! Example 2 Substitute. Solve for k. Use 12 for k. Substitute 75 for P. Solve for s. P = ks Method 1 Find k. 18 = k(1.5) 12 = k Write the variation function. 75 = 12s 6.25 s P = 12s

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Holt Algebra Variation Functions Substitute. Find the cross products. Method 2 Use a proportion. P1P1 = s1s1 s2s2 P2P2 18 = 1.5s 75 18s = Solve for s = s Check It Out! Example 2 Continued

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Holt Algebra Variation Functions A joint variation is a relationship among three variables that can be written in the form y = kxz, where k is the constant of variation. For the equation y = kxz, y varies jointly as x and z. The phrases y varies directly as x and y is directly proportional to x have the same meaning. Reading Math

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Holt Algebra Variation Functions The volume V of a cone varies jointly as the area of the base B and the height h, and V = 12 ft 3 when B = 9 ft 3 and h = 4 ft. Find b when V = 24 ft 3 and h = 9 ft. Example 3: Solving Joint Variation Problems Step 1 Find k. The base is 8 ft 2. Substitute. Solve for k. V = kBh 12 = k(9)(4) = k 1 3 Substitute. Solve for B. Step 2 Use the variation function. 24 = B(9) 8 = B V = Bh Use for k. 1 3

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Holt Algebra Variation Functions Check It Out! Example 3 The lateral surface area L of a cone varies jointly as the area of the base radius r and the slant height l, and L = 63 m 2 when r = 3.5 m and l = 18 m. Find r to the nearest tenth when L = 8 m 2 and l = 5 m. Step 1 Find k. Substitute. Solve for k. L = krl 63 = k(3.5)(18) = k Substitute. Solve for r. Step 2 Use the variation function. 8 = r(5) 1.6 = r L = rl Use for k.

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Holt Algebra Variation Functions A third type of variation describes a situation in which one quantity increases and the other decreases. For example, the table shows that the time needed to drive 600 miles decreases as speed increases. This type of variation is an inverse variation. An inverse variation is a relationship between two variables x and y that can be written in the form y =, where k 0. For the equation y =, y varies inversely as x. k x k x

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Holt Algebra Variation Functions Given: y varies inversely as x, and y = 4 when x = 5. Write and graph the inverse variation function. Example 4: Writing and Graphing Inverse Variation Substitute 4 for y and 5 for x. Solve for k. 4 =4 = k = 20 y = k 5 y varies inversely as x. k x Write the variation formula. y =y = 20 x

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Holt Algebra Variation Functions To graph, make a table of values for both positive and negative values of x. Plot the points, and connect them with two smooth curves. Because division by 0 is undefined, the function is undefined when x = 0. Example 4 Continued xy –2–10 –4–5 –6– –8– xy

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Holt Algebra Variation Functions Check It Out! Example 4 Given: y varies inversely as x, and y = 4 when x = 10. Write and graph the inverse variation function. Substitute 4 for y and 10 for x. Solve for k. 4 =4 = k = 40 y = k 10 y varies inversely as x. k x Write the variation formula. y =y = 40 x

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Holt Algebra Variation Functions Check It Out! Example 4 Continued To graph, make a table of values for both positive and negative values of x. Plot the points, and connect them with two smooth curves. Because division by 0 is undefined, the function is undefined when x = 0. xy –2–20 –4–10 –6– –8–5 xy

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Holt Algebra Variation Functions When you want to find specific values in an inverse variation problem, you can solve for k and then use substitution or you can use the equation derived below.

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Holt Algebra Variation Functions Example 5: Sports Application The time t needed to complete a certain race varies inversely as the runners average speed s. If a runner with an average speed of 8.82 mi/h completes the race in 2.97 h, what is the average speed of a runner who completes the race in 3.5 h? Method 1 Find k. Substitute. Solve for k = k = t = k 8.82 k s Use for k. t = s Substitute 3.5 for t. 3.5 = s Solve for s. s 7.48

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Holt Algebra Variation Functions Method Use t 1 s 1 = t 2 s 2. t 1 s 1 = t 2 s s = 3.5s (2.97)(8.82) = 3.5s Simplify. Solve for s. Substitute. So the average speed of a runner who completes the race in 3.5 h is approximately 7.48 mi/h. Example 5 Continued

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Holt Algebra Variation Functions The time t that it takes for a group of volunteers to construct a house varies inversely as the number of volunteers v. If 20 volunteers can build a house in 62.5 working hours, how many working hours would it take 15 volunteers to build a house? Method 1 Find k. Substitute. Solve for k = k = 1250 t = k 20 k v Use 1250 for k. t = 1250 v Substitute 15 for v. t = Solve for t. t 83 Check It Out! Example 5 1 3

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Holt Algebra Variation Functions Method 2 Use t 1 v 1 = t 2 v 2. t 1 v 1 = t 2 v 2 83 t 1250 = 15t (62.5)(20) = 15t Simplify. Solve for t. Substitute So the number of working hours it would take 15 volunteers to build a house is approximately hours. 1 3 Check It Out! Example 5 Continued

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Holt Algebra Variation Functions You can use algebra to rewrite variation functions in terms of k. Notice that in direct variation, the ratio of the two quantities is constant. In inverse variation, the product of the two quantities is constant.

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Holt Algebra Variation Functions Example 6: Identifying Direct and Inverse Variation Determine whether each data set represents a direct variation, an inverse variation, or neither. A. x y840.5 In each case xy = 52. The product is constant, so this represents an inverse variation. B. x5812 y In each case = 6. The ratio is constant, so this represents a direct variation. y x

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Holt Algebra Variation Functions Determine whether each data set represents a direct variation, an inverse variation, or neither. C. x368 y51421 Since xy and are not constant, this is neither a direct variation nor an inverse variation. y x Example 6: Identifying Direct and Inverse Variation

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Holt Algebra Variation Functions Determine whether each data set represents a direct variation, an inverse variation, or neither. 6a. x y1239 In each case xy = 45. The ratio is constant, so this represents an inverse variation. Check It Out! Example 6 6b. x14026 y In each case = 0.2. The ratio is constant, so this represents a direct variation. y x

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Holt Algebra Variation Functions A combined variation is a relationship that contains both direct and inverse variation. Quantities that vary directly appear in the numerator, and quantities that vary inversely appear in the denominator.

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Holt Algebra Variation Functions The change in temperature of an aluminum wire varies inversely as its mass m and directly as the amount of heat energy E transferred. The temperature of an aluminum wire with a mass of 0.1 kg rises 5°C when 450 joules (J) of heat energy are transferred to it. How much heat energy must be transferred to an aluminum wire with a mass of 0.2 kg raise its temperature 20°C? Example 7: Chemistry Application

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Holt Algebra Variation Functions Step 1 Find k. The amount of heat energy that must be transferred is 3600 joules (J). Substitute. Solve for k. ΔT = 5 =5 = = k Substitute. Solve for E. Step 2 Use the variation function = E Use for k kE m Combined variation k(450) 0.1 ΔT = E 900m 20 = E 900(0.2) Example 7 Continued

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Holt Algebra Variation Functions Check It Out! Example 7 The volume V of a gas varies inversely as the pressure P and directly as the temperature T. A certain gas has a volume of 10 liters (L), a temperature of 300 kelvins (K), and a pressure of 1.5 atmospheres (atm). If the gas is heated to 400K, and has a pressure of 1 atm, what is its volume?

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Holt Algebra Variation Functions Step 1 Find k. The new volume will be 20 L. Substitute. Solve for k. V = 10 = 0.05 = k Substitute. Solve for V. Step 2 Use the variation function. V = 20 Use 0.05 for k. Combined variation k(300) 1.5 V = 0.05T P V = 0.05(400) (1) kT P Check It Out! Example 7

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Holt Algebra Variation Functions Lesson Quiz: Part I 1. The volume V of a pyramid varies jointly as the area of the base B and the height h, and V = 24 ft 3 when B = 12 ft 2 and h = 6 ft. Find B when V = 54 ft 3 and h = 9 ft. 18 ft 2 2. The cost per person c of chartering a tour bus varies inversely as the number of passengers n. If it costs $22.50 per person to charter a bus for 20 passengers, how much will it cost per person to charter a bus for 36 passengers? $12.50

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Holt Algebra Variation Functions Lesson Quiz: Part II 3. The pressure P of a gas varies inversely as its volume V and directly as the temperature T. A certain gas has a pressure of 2.7 atm, a volume of 3.6 L, and a temperature of 324 K. If the volume of the gas is kept constant and the temperature is increased to 396 K, what will the new pressure be? 3.3 atm

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